Lawsuits and 1 Corinthians 6

The cultural setting of Scripture is always important. This is particularly true when the authors of Scripture suggested/required particular applications to their original audiences. In the case of Paul's teachings about lawsuits in 1 Corinthians 6, Paul's argument seems to have included at least two foundational ideas.

First, Paul believed that Christians ought to be better equipped than secular courts to solve disputes between Christians (1 Cor. 6:5). Because of this, he exhorted the church to appoint arbitrators or mediators to work out disputes between Christians. I think he was assuming (rightly, of course) that secular law doesn't always line up with biblical values, and therefore that the church better equipped to reach judgments that line up with God's standards. I don't think he had criminal matters in mind here (cf. Rom. 13), or other matters related directly to the state (taxes, licensing, regulations, etc.). But many other types of legal matters might fall under this command (civil, family, etc.).

Second, Paul also believed it was a matter of honor for Christians not to appeal to the civil government to solve their internal problems (1 Cor. 6:6). Preferring state solutions over church solutions suggests that Christians don't really believe that we are first and foremost citizens of God's kingdom, and only secondarily citizens of other nations. It implies that we prefer the world's authority and justice over God's authority and justice. Paul even went so far as to say that if it turned out that the church was incompetent (contrary to his first assumption, mentioned above), it was better to suffer injustice at the hands of the church than to insult God by seeking civil justice.

In the case you mention, it sounds like the Christian parties don't prefer civil justice over biblical justice, and don't believe that state courts are more competent. Rather, they simply want to compel a third party (the insurance company) to approach the negotiating table. With respect to the insurance company, it's reasonable and biblical to involve the state courts. One way to do that would be to have the injured party sue the insured party as a means of obtaining an insurance settlement. In my judgment, neither the motive for using the state court, nor the service provided by the state court, would violate the principles Paul taught in 1 Corinthians 6. Ideally, the Christian parties would agree to be bound by the church's judgment in the event that a settlement could not be obtained from the insurance company.

That being said, there may also be other solutions. For example, the church could determine the insured's liability and render a decision. If the church found the insured liable, and if the insurance company didn't agree, the insured could sue his own insurance company in an attempt to obtain a settlement for the injured party. I'm not an attorney, and I don't even play one on TV, so don't take that as legal advice. But there are often a variety of biblical ways to involve state courts - ways that honor both the authority God has vested in the church and the authority he has vested in the civil government. In no case would I conclude that Scripture always forbids the use of state courts when Christians are involved in more than one side of a dispute.

Hope this helps.